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Ore Sets in Enveloping Algebras Compositio Mathematica, Tome 53, No 3 (1984), P COMPOSITIO MATHEMATICA KENNETH A. BROWN Ore sets in enveloping algebras Compositio Mathematica, tome 53, no 3 (1984), p. 347-367 <http://www.numdam.org/item?id=CM_1984__53_3_347_0> © Foundation Compositio Mathematica, 1984, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Compositio Mathematica 53 (1984) 347-367. © 1984 Martinus Nijhoff Publishers, Dordrecht. Printed in The Netherlands. ORE SETS IN ENVELOPING ALGEBRAS Kenneth A. Brown 1. Introduction Let g be a finite dimensional Lie algebra over C, with enveloping algebra aJ/(g). In the study of u(g) and its representation theory, localisation has proved to be the most useful weapon from the armoury of non-com- mutative ring theory. In this paper we sharpen that weapon by describ- ing, for a prime ideal P of the enveloping algebra of a solvable subalge- bra p of g, the largest Ore set 5°(P ) contained in Lu(p)(P) = u(p)BP, (5.3) and (7.1). (A subset 9-’of a ring R is a right Ore set, or satisfies the right Ore condition if for all s E 9-’and r E R there exist dE 9-’and u E R such that su = rd. In this case one can localise at p to form the ring Rp = {ac-1: a ~ R, c ~ p}, where denotes images in R/I, I = {r ~ R: rs = 0, s E p}. Similar definitions apply on the left. An Ore set is a right and left Ore set.) In §§2, 5 and 6, we deal with the case p = g. In §2 we derive a necessary condition on 9-’(P) in terms of certain subsemigroups of (g/[g, g])* arising from the Jordan-Hôlder values of g (2.3). In §5 it is shown that the largest subset W(P) satisfying this necessary condition is, in fact, an Ore set (5.3); analogous results are obtained for the maximal right, and left, Ore sets, p’(P) and ’p(P), in l(P). Sections 3 and 4 contain results needed for the proof of (5.3) which may be of indepen- dent interest to non-commutative ring theorists. The structure of the ring aJ/ (g ),9’( P) is examined in §6, (again here g = p is solvable). We prove that u(g)p(P) is a Noetherian domain with only countably many maximal right or left ideals, each of which is a twosided ideal generated by the image of P under a suitable winding automorphism of OJ/(g). Let M be one such ideal. Then u(g)p(P)/M is isomorphic to the division ring of quotients of &#x26;(g)IP, and M has a regular normalising set of t generators, (see (5.3) for the definition), where t, the height of P and of M, is the global dimension and the Krull dimension of u(g)p(P). Every primitive ideal of u(g)p(P) is maximal, u(g)p(P) has stable range 1, (see (6.2) for the definition), and every projective module over u(g)p(P) is free. When P is localisable, that is, when p(P) = b(P), then of course u(g)p(P) is a "local ring". In view of the above proper- 347 348 ties, one might feel that the tranditional generalisation of the definition of "local ring" from the commutative to the non-commutative context has been too restrictive; we discuss this further in (6.3). By localising at p’(P) instead of p(P), one can obtain interesting right Noetherian left Ore domains; see (6.4). In the next section we consider the general case where p ç g. A sufficient condition for an Ore subset of u(p) to be Ore in u(g) was obtained in [5, 4.5]. Theorem 7.1 shows that, with the addition of a mild technical assumption on the Ore sets in question, this condition is also necessary. One can thus describe the maximal Ore subset of lu(p)(P) in u(g), for a prime ideal P of u(p). In Theorem 8.2 we specialise this description to the case where p is a Borel subalgebra b of a semisimple Lie algebra g. The statement is particularly neat where P is the annihila- tor of a 1-dimensional *(b)-module, say P = l(C03BC) where 03BC ~ h*, h a Cartan subalgebra in b ; here, Y(P), the largest Ore subset of lu(b)(P) in u(g), is ~{bu(b)(l(C03BB)): À E JL + Z R}, where R is the set of roots of g relative to h, (8.2(iii)). The rest of §8 consists of some observations on the structure of u(g)p(P) for this choice of g and P. A word about coefficient fields. While we have assumed throughout that Lie algebras are complex, much of [7], and hence, much of this paper can be routinely generalised to a completely solvable Lie algebra over an arbitrary field. The results on Ore sets described here can be applied to enveloping algebras over other coefficient fields using the following results. 1.1. LEMMA: Let K c F be fields and g a Lie algebra, finite dimensional over K. Then a subset!/ of *(g) satisfies the Ore condition if and only if Y satisfies the Ore condition in u(g) ~KF. 1.2. LEMMA: Let K ~ F be fields and let U be a K-algebra with the ascending chain condition on ideals. Let 9 be a set of completely prime ideals of U ~KF. Suppose that |P| K and that Y= ~ b(P) is an Ore POEP set in U 0 F. Then p~U is an Ore set in U. The proof of 1.1 is an easy exercise; the proof of 1.2 is sketched in 4.7. In the hope of stimulating applications of this work to the further study of enveloping algebras, we have tried to make this paper readable by the non-specialist in Noetherian ring theory; but it may be as well to note that the approach to localisation described here originates in work of A.V. Jategaonkar, recent accounts of which may be found in [12], [14]. The fact that Jategonkar’s theory, much of which applied originally only to FBN rings, could be made to apply to certain other classes of Noetherian rings, was observed independently by Jategaonkar and the author [13], [6]. The key idea from this theory which underlies the present paper is that of a prime link, which is implicit in 2.1. 349 The nature of the " prime links" for enveloping algebras of complex solvable Lie algebras was determined in [7], but for this paper we in effect need to know only the connected components of the graph of links, rather than the exact form of the edge set. It is an interesting but apparently difficult question for Noetherian ring theory, whether the Ore sets obtained in this paper, as intersections determined by the graph of links between primes, exist in a more general, more abstract, setting. The only other results obtained to date in this direction apply to certain P.I. rings [18]. * 2. Maximal possible Ore set In this section g denotes a finite dimensional complex solvable Lie algebra. Let P be a prime ideal of q¡ (g). Our target is a necessary condition (2.3) for a subset of b(P) to satisfy a one-sided Ore condition. (We denote the set of regular elements modulo an ideal I of a ring S by b(I), or bS(I).) 2.1. LEMMA: Let R be a Noetherian ring with prime ideals P and Q, and let p~ b(P) be a left Ore set. Let A c B be ideals of R with l(B/A) = P and r(B/A) = Q. Then9’ç b(Q). PROOF: The case where B/A is R/P- and R-Q-torsion free is given by [14, 2.3], (or see [7, 1.4]). The required result reduces easily to this case, noting that if I is a prime containing Q, and p ~ b(I), then p ~ b(Q). = 2.2. Let 0 go c ... c gn = g be a composition series of g, so gl ~g and E gilg, -1 has dimension one, 1 ~ i ~ n, [4, 6.6]. Let À, g * be the eigen- value of ad g on gi/gi-1, 1 :s; i ~ n, so 03BB1, À n are the Jordan-Hôlder values of g. fix For a ideal P of let For i = 1, ..., n, x, ~ gl Bgl-1. prime u(g), Pl = P~u(gl), 0 ~ i ~ n, and Yc {1, ... , n ) consist of those integers for which Pl ~ Pl -1 1 &#x26; d (g,). For i ~ p, let m; be the minimal degree in x, of an element of PlBPl-1u(gi). Let the subsemigroups of (g/[g, g])* generated by {ml03BBl: i ~ p}, {-mi03BBl: i ~p}, and ( ± mixi: i ~ p} be denoted by L(P), R ( P ) and Q( P) respectively. (This notation is apparently defective, since L ( P ), etc. seem to depend on our choice of composition series for g. But, as will become apparent from 5.3, these subsemigroups are independent of this choice.) Recall that for JL E (g/[g, g])* the winding automorphism r of u(g) is defined by T(x) = x + p(x) for x E g, [4, 10.2].
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